cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A110129 Central coefficients of a scaled Legendre triangle.

Original entry on oeis.org

1, 2, 22, 504, 16966, 752800, 41492284, 2734083968, 209681631814, 18348172005888, 1804161160185748, 196945525458761728, 23633625832975567644, 3092337510752711057408, 438161926888980929318584, 66838962347916069718425600, 10921339483491720513675526726, 1903085098399657078831752282112
Offset: 0

Views

Author

Paul Barry, Jul 13 2005

Keywords

Comments

Central coefficients of triangle A110124.

Links

Crossrefs

Cf. A008556, A110124, A349077, A349113, A349114, A349115.

Programs

  • Maple
    a:= n-> LegendreP(n$2)*2^n:
seq(a(n), n=0..17);  # Alois P. Heinz, Nov 17 2024
  • Mathematica
    Table[2^n LegendreP[n,n],{n,0,20}] (* Harvey P. Dale, Nov 28 2012 *)
  • PARI
    a(n)=pollegendre(n,n)<Charles R Greathouse IV, Mar 19 2017

Formula

a(n) = 2^n*LegendreP(n, n).
a(n) = Sum_{j=0..floor(n/2)} (-1)^j*C(n, j)*C(2*n-2*j, n)*n^(n-2*j).
a(n) ~ 2^(2*n) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Nov 07 2021

A349113 a(n) = 8^n * P(3*n, n), where P(n, x) is n-th Legendre polynomial.

Original entry on oeis.org

1, 8, 40636, 748832256, 37759888297756, 4086692369433395200, 815254385427670754825764, 270587150855247020644760551424, 138859707622050969870951620062449436, 104286590422721059977069662227099300134912, 109828573459404650800550127862919905133973562480
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 08 2021

Keywords

Comments

In general, for k>=1, P(k*n, n) ~ 2^(k*n) * n^(k*n) / sqrt(k*Pi*n).

Links

Crossrefs

Cf. A008316, A110129, A349077, A349115.

Programs

  • Mathematica
    Table[8^n*LegendreP[3*n, n], {n, 0, 12}]
  • PARI
    a(n) = 8^n*pollegendre(3*n, n); \\ Michel Marcus, Nov 08 2021

Formula

a(n) ~ 2^(6*n) * n^(3*n - 1/2) / sqrt(3*Pi).

A349114 a(n) = 4^n * P(n, 2*n), where P(n, x) is n-th Legendre polynomial.

Original entry on oeis.org

1, 8, 376, 33984, 4526176, 797459200, 174910868224, 45926958135296, 14047764722238976, 4905641267399503872, 1925859774286175997952, 839619968812285810868224, 402496047174560754869846016, 210424519428145503482634174464, 119148510992477432889126160826368
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 08 2021

Keywords

Links

Crossrefs

Cf. A008316, A110129, A349077, A349115.

Programs

  • Mathematica
    Table[4^n*LegendreP[n, 2*n], {n, 0, 16}]
  • PARI
    a(n) = 4^n*pollegendre(n, 2*n); \\ Michel Marcus, Nov 08 2021

Formula

a(n) ~ 2^(4*n) * n^(n - 1/2) / sqrt(Pi).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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